This article is devoted to a deep study of the Roper-Suffridge extension operator with special geometric properties. First, we prove that the Roper-Suffridge extension operator preserves ϵ starlikeness on the open unit ball of a complex Banach space ℂ × X, where X is a complex Banach space. This result includes many known results. Secondly, by introducing a new class of almost boundary starlike mappings of order α on the unit ball Bn of ℂn, we prove that the Roper-Suffridge extension operator preserves almost boundary starlikeness of order α on Bn. Finally, we propose some problems.

中文翻译：

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ε 星状和边界星状映射的广义 Roper-Suffridge 算子

本文致力于深入研究具有特殊几何性质的 Roper-Suffridge 可拓算子。首先，我们证明了 Roper-Suffridge 扩展算子在复 Banach 空间 ℂ × X 的开单位球上保留了 ϵ 星状，其中 X 是复 Banach 空间。这个结果包括许多已知的结果。其次，通过在 ℂn 的单位球 Bn 上引入一类新的 α 阶近似边界星状映射，我们证明了 Roper-Suffridge 扩展算子在 Bn 上保持了 α 阶近似边界星状映射。最后，我们提出一些问题。